Theorem rngohommul 35263

Description: Ring homomorphisms preserve multiplication. (Contributed by Jeff Madsen, 3-Jan-2011.)

Hypotheses

Ref	Expression
rnghommul.1	⊢ 𝐺 = (1st ‘𝑅)
rnghommul.2	⊢ 𝑋 = ran 𝐺
rnghommul.3	⊢ 𝐻 = (2nd ‘𝑅)
rnghommul.4	⊢ 𝐾 = (2nd ‘𝑆)

Assertion

Ref	Expression
rngohommul	⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))

Proof of Theorem rngohommul

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnghommul.1	. . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅)
2		rnghommul.3	. . . . . . 7 ⊢ 𝐻 = (2nd ‘𝑅)
3		rnghommul.2	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
4		eqid 2821	. . . . . . 7 ⊢ (GId‘𝐻) = (GId‘𝐻)
5		eqid 2821	. . . . . . 7 ⊢ (1st ‘𝑆) = (1st ‘𝑆)
6		rnghommul.4	. . . . . . 7 ⊢ 𝐾 = (2nd ‘𝑆)
7		eqid 2821	. . . . . . 7 ⊢ ran (1st ‘𝑆) = ran (1st ‘𝑆)
8		eqid 2821	. . . . . . 7 ⊢ (GId‘𝐾) = (GId‘𝐾)
9	1, 2, 3, 4, 5, 6, 7, 8	isrngohom 35258	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ (𝐹:𝑋⟶ran (1st ‘𝑆) ∧ (𝐹‘(GId‘𝐻)) = (GId‘𝐾) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))))
10	9	biimpa 479	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:𝑋⟶ran (1st ‘𝑆) ∧ (𝐹‘(GId‘𝐻)) = (GId‘𝐾) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))))
11	10	simp3d 1140	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))
12	11	3impa 1106	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))
13		simpr 487	. . . 4 ⊢ (((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))) → (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))
14	13	2ralimi 3161	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹‘𝑥)(1st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))
15	12, 14	syl 17	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))
16		fvoveq1 7179	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥𝐻𝑦)) = (𝐹‘(𝐴𝐻𝑦)))
17		fveq2 6670	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
18	17	oveq1d 7171	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥)𝐾(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝑦)))
19	16, 18	eqeq12d 2837	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)) ↔ (𝐹‘(𝐴𝐻𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝑦))))
20		oveq2 7164	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
21	20	fveq2d 6674	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝐴𝐻𝑦)) = (𝐹‘(𝐴𝐻𝐵)))
22		fveq2 6670	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
23	22	oveq2d 7172	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴)𝐾(𝐹‘𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))
24	21, 23	eqeq12d 2837	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐹‘(𝐴𝐻𝑦)) = ((𝐹‘𝐴)𝐾(𝐹‘𝑦)) ↔ (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵))))
25	19, 24	rspc2v 3633	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵))))
26	15, 25	mpan9 509	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹‘𝐴)𝐾(𝐹‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ran crn 5556  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  1^st c1st 7687  2^nd c2nd 7688  GIdcgi 28267  RingOpscrngo 35187   RngHom crnghom 35253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-map 8408  df-rngohom 35256

This theorem is referenced by:  rngohomco  35267  rngoisocnv  35274  crngohomfo  35299  keridl  35325